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# Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i) $${{1} \over {4}} , -1$$
(ii) $$\sqrt{2}, {{1}\over{3}}$$
(iii) $$0, \sqrt{5}$$
(iv) $$1,1$$
(v) $${{-1} \over {4}},{{1} \over {4}}$$
(vi) $$4,1$$

(i) $${{1} \over {4}} , -1$$
$$\Rightarrow$$Sum of the zeroes = $${{1} \over {4}}$$ = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = -1 = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$$$x^2 - {{1} \over {4}}x + (-1)$$ => $$4x^2 - x - 4$$

(ii) $$\sqrt{2} , {{1}\over{3}}$$
$$\Rightarrow$$Sum of the zeroes = $$\sqrt{2}$$ = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = $${{1} \over {3}}$$ = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$$$x^2 - \sqrt{2} x + {{1} \over {3}}$$ $$\Rightarrow$$ $$3x^2 - 3\sqrt{2}x + 1$$

(iii) $$0 , \sqrt{5}$$
$$\Rightarrow$$Sum of the zeroes = 0 = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = $$\sqrt{5}$$ = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$$$x^2 - 0x + ( \sqrt{5})$$ => $$x^2 + \sqrt{5}$$

(iv) $$1 , 1$$
$$\Rightarrow$$Sum of the zeroes = 1 = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = 1 = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$ $$x^2 - 1x + 1$$

(v) $${{-1} \over {4}} , {{1} \over {4}}$$
$$\Rightarrow$$ Sum of the zeroes = $${{-1} \over {4}}$$ = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = $${{1} \over {4}}$$ = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$ $$x^2 - {{-1} \over {4}}x + {{1} \over {4}}$$ $$\Rightarrow$$ $$4x^2 + x + 1$$

(vi) $$4 , 1$$
$$\Rightarrow$$ Sum of the zeroes = 4 = $${{-b}\over {a}}$$
$$\Rightarrow$$ Product of the zeroes = 1 = $${{c}\over {a}}$$
Polynomial can be formed by : $$x^2$$ - (sum of zeroes)x + (product of zeroes)
$$\Rightarrow$$ $$x^2 - 4x + 1$$